Nonclassified activity of CML RFNC-VNIITF Oleg V. Diyankov Head of Computational MHD Laboratory.

Nonclassified activity of CML RFNC-VNIITF

Oleg V. Diyankov

Head of Computational MHD Laboratory

Russian Federal Nuclear Center-All-Russian Institute for Technical Physics

Snezhinsk, Chelyabinsk region (the Urals), Russia

About 9000 employees, among which there are scientists: physicists, chemists, mathematicians; designers, engineers, etc.

RFNC has many divisions. We’re representing the division of theoretical physics and applied mathematics.

Division of Theoretical Physics and applied mathematics

260 scientists in computational mathematics and theoretical physics.

Computational MHD lab was created on the 1st of April 1996.

14 scientists are working in it.

The picture of the laboratory

The main directions of the work 2.5D MHD code 2D Irregular Grid for Mathematical Modeling Linear Solvers for Flows in Porous Media Modeling Difference Schemes for Hyperbolic Systems Treatment 3D Elastic-Plastic Modeling of Processes of Ceramics

Formation 3D Gas Dynamic Code for Instability Investigation Development of Special Software Tools for CERN

2.5D MHD Code

The Physical Model Realized in the Code The Application of the Code to Laser Beam

Interaction with the Matter Modeling The Application of the Code to Liner

Magnetic Compression Modeling The Application of the Code to the Plasma

Channel Formation Modeling

2½D MHD MAG Code 2½D MHD MAG Code

O.V.Diyankov, I.V.Glazyrin,

S.V.Koshelev, I.V.Krasnogorov, A.N.Slesareva, O.G.Kotova

Russian Federal Nuclear Center - VNIITFP.O.Box 245, Snezhinsk, Chelyabinsk Region, Russia

The main goal of MAG code creation was the necessity of modeling of hot dense plasmas in magnetic field.

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The system of equations used in MAG code is determined by Braginskii model for one-temperature case:

The MAG Code Model:The MAG Code Model:

Let us assign indices x and y in the case of the axial symmetry to the r- and z- components of vector variables and r- and z- components of independent spatial variables correspondently, and index z to the - component of vector variables.

Then we receive a basic system of equations, which depends on the parameter of symmetry ( = 0 - the plane symmetry, = 1 - the symmetry is axial).

The equations have been written in arbitrary moving coordinate system, and then splitting into two systems has been performed. The diffusive terms have been splitted to a separate system of equations, the remained terms produce a quasilinear hyperbolic system.

Two equations for x- and y- components of magnetic field was written in form of an z-component vector potential A:

The first system is a hyperbolic one and it describes the ideal MHD flows in arbitrary moving coordinate system.

The second one is a diffusive system of equations. It includes the equations for energy, z (or ) – component of magnetic field and z (or ) component of vector potential.

Details of Numeric:Details of Numeric:

x

AxB

y

AxB yx

;

1. Gas dynamics conditions:

• applied pressure: P|b= f(t), where P|b means the pressure at the corresponding boundary, f(t) is a given time dependent function;

• rigid wall: un|b = 0, where un is a normal to boundary

component of mass velocity;

• piston: un|b = f(t);

Boundary conditions:Boundary conditions:

2. Conditions for heat conductivity:

),( tTfTTbbbn

They are: given temperature, given heat flux, heat flux as a function of temperature. These conditions may be written in the form:

Here, T|b is the temperature at the corresponding boundary, , are numerical parameters (=0 and =1 for given temperature and =1 and =0 for heat flux), f(T|b, t) is a given function. nT|b is a normal to the boundary component of the temperature gradient.


3. Conditions for magnetic field


• symmetry: Bz /n=0, B /n=0, Bn =0 ;

• conducting wall: Bz=0, B=0, Bn =0 ;

• axis: Bz=0, B=0, Bn =0, where B is used for the axial symmetry and Bz for the plane one;

• given current: Bz=2j /c – plane symmetry, B= 2I /cr – axial symmetry,

where Bn is the normal to the boundary component of the magnetic field, j - current density, I is the whole current in z - direction, r - upper radius.

Algorithms of mesh reconstruction:Algorithms of mesh reconstruction:

Lagrange (no mesh reconstruction) Euler (grid nodes are returned to original positions at the

n-th time step) Local (only eight neighbor nodes are used for new node

coordinates determination) Algebraic (new nodes coordinates are calculated by

bilinear interpolation of boundaries nodes coordinates in mathematic coordinate system)

Poison equation solution is used for new coordinates determination

System of equations in arbitrary moving coordinates I:

System of equations in arbitrary moving coordinates I:

Equation of continuity:

Euler equations for velocity components:

x component:

y component:

z component:

System of equations in arbitrary moving coordinates II:System of equations in arbitrary moving coordinates II:

Equation for A:

Equation for z component of magnetic field:

Equation for total energy:

Equation for the square root of the metric tensor:

Here uk is a projection of mass velocity vector to k-th vector of a local basis, uk=uk, where k is a covariant local basis vector, vk,Bk are determined in the same way, v - coordinate system velocity, g is a determinant of metric tensor with covariant component gij,

k - mathematical coordinates.

System of equations in arbitrary moving coordinates III:System of equations in arbitrary moving coordinates III:

= const. - equation of continuity,g = const. - equation for determinant of metric tensor,u = const. - Euler equation for velocity.

Equation for A:

Equation for Bz:

Equation for energy:

Difference scheme for single diffusive equation:Difference scheme for single diffusive equation:

Where: kk

g

D

Diffusive equations system:Diffusive equations system:

Difference sheme for diffusive equations system:Difference sheme for diffusive equations system:

Z-pinch simulationZ-pinch simulation

Anod

Cathod

Experimental Setup

If plasma liner implosion is used as plasma radiation source one needs to receive the uniform plasma column. The ideal configuration for such type of radiation source could be an annular pinch (see Fig.1, left). The plasma, imploded from large radius, reaches the axis and the efficient radiator is formed. But this scheme is unsuitable because of MHD instabilities which present the great danger to uniformity of the liner implosion. So the compression achieved in corresponding experiments is substantially lower than one predicted by one-dimensional (1D) MHD calculations.

Simulation

Hollow gas puff simulation:Hollow gas puff simulation:

Density [g/cc]

Magnetic field, -component

[10 MGs]

Pinhole image reconstruction throw steady equation of radiation transfer resolving

Pinhole image reconstruction throw steady equation of radiation transfer resolving

IS

l

I

dSeIeIl d sd l

ll

0

00

E n erg y a d sorb ed b y film :

0

dISFdE

W h ere:( ) - p in h o le filte r fu n ction

SF

( ) - film sen s ib ility fu n ctio nG e om etr ic p ar am eters

The equation of radiation transfer along a ray has the following form:

Pinhole Image Reconstruction on Simulation ofGas Puff Implosion (I)

Pinhole Image Reconstruction on Simulation ofGas Puff Implosion (I)

Pinhole Im age

Density[g/cc]

Temperature[keV]

193.5 ns

180 ns

Pinhole Image Reconstruction on Simulation ofGas Puff Implosion (II)

Pinhole Image Reconstruction on Simulation ofGas Puff Implosion (II)

196.5 ns

204.5 ns

One Shell Gas PuffMass 100 g/cmR adius 2.2 cm 10% random initial density disturbance

Pinhole Im age

Density[g/cc]

Temperature[keV]

Simulation parameters:

Current:1.7 MA

100 ns rise time

Laser produced plasma jet expansion into vacuum

Laser produced plasma jet expansion into vacuum

Temperature evolution, time = 0.1, 0.3, 0.5, 0.8

ns. Energy of laser pulse is4 kJ/cm2. Triangle pulse

with duration time of 1 ns. Focal spot 50 m.

Laser produced plasma jet expansion into background plasma

Laser produced plasma jet expansion into background plasma

Temperature evolution. time = 0.1, 0.3, 0.5, 0.8

ns. Energy of laser pulse

4 kJ/cm2. Triangle pulse shape with duration time

of 1 ns. Focal spot 50 m.Background plasma

density was 10-6 g/cm3

Experimental SetupExperimental Setup

Laser B eam

G as JetVa lve

P la sm a is c rea ted du e to in te rac tion o f la se r p u lse w ith a g a s p u ff ta rge t.

In itia l D en s ity P ro file

L a se r P aram ete rs :N d :g la ss 1 n s

1 0 J5 0 m

la se r sy s tem p ro d uc ing p u lse s w ith e n erg y u p to a n d fo ca l rad iu s m

Plasma Channel Formation. Density Evolution.Plasma Channel Formation. Density Evolution.

Tim e: ns0.4 Tim e: ns0.6


Plasma Channel Formation. Temperature Evolution.Plasma Channel Formation. Temperature Evolution.



The plasma is heated up to high temperature (temperature reaches

450 eV).

A motion of the SW should be spherical but the SW dynamics

propagating along the laser pulse direction differs from the perpendicular SW one.

As the velocity of SWs is approximately equal to 107 cm/sec,

the perpendicular SW leaves the region of the laser absorption (the

focal radius) after 0.3-0.5 ns.

2D Irregular Grid for Mathematical Modeling

Gas Dynamics Poisson Equation Solver Maxwell Equation Solver Heat Transfer

Oleg V. Diyankov

Sergei S. Kotegov

Vladislav Yu. Pravilnikov

Yuri Yu. Kuznetsov

Aleksey A. Nadolskiy

RFNC – ARITPh

supported by LLNL grant B329117

IGM CodeIGM Code

Overview

The IGM code was created to perform 2D flows modeling. The main feature of the code is the possibility of large deformations accounting.

3 physical processes are taken into account now: gas dynamics, heat conduction and Poisson equation.

The main advantage before well-known finite element codes is the possibility of arbitrary deformations description.

Features & Benefits

The flow region is initially covered by a set of Voronoi cells, and then at each time step the grid is reconstructed.

This allows neighbor points to move free in any direction, so they may move very far from each other.

The GUI interface for the IGM code (it is called CELLS) allows to put in initial data (geometry, matters, initial distribution of the values), and to look through the received results.

Applications

Plasma physics (instability study, laser produced plasma, and so on).

High velocity impact. Heat transfer. Electrostatic fields.

Voronoi diagram

Benefits: Local orthogonality Local uniformity

Gas-dynamic equations with heat conductivity

),(

2

u

u

gradu

G)(

0

2

pp

E

x

Tkdivgpt

gE

gpt

gu

t

g

ii

ii

Splitting according to physical processes

u

0u

G)(

0

x

gpt

gE

gpt

gu

t

g

ii

ii

Tdivt

t

t

grad

0u

0

and

Difference scheme

k

j

Ei

ni

ni

k

j

ui

n

iyn

iy

k

j

ui

n

ixn

ix

ni

ni

j

y

j

x

j

SEmEm

Sumum

Sumum

mm

1

1

1

1

1

1

1

corresponding fluxes

l

ijeen

ijee

ee

ijejjii

Ej

yeeyij

eejiuj

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eejiuj

ucuc

c

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lcncppS

y

x

l

luu

2

1

n

nuu

2

1

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l

luu

2

1

n

nuu

2

1

2

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l

luu

2

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n

nuu

2

1

2

1

ij

ij

coordinates

k

jee

v

l

k

j ee

e

v

ln

nn

cKK

K

c

p

K

K

1

11

n2

n1

2u

rr

Equation of heat conductivity

0S div

t

TCv

n1

1

111

k

j

ni

nj

ne

i

ni

nin

iv TTV

TTC

difference scheme:

tfTTn

boundary condition:

Poisson equation)(rfdiv

Integrate over Voronoi cell

SS

dSxrfdln

x )(

Sxrfnn

xxiiij

ij

ji

i

)(

2

where

2

2

1

1

GG

iG

G

ij

ij

ij

n

dn

rrn

Electric field strength

jkli

i

SS

lnS

E

dSdSE

E

i

1

2,

2

2

kiik

jiij

ikijl

where

The development of Raleigh-Taylor instability

The grid at the last moment for RT instability test.

High velocity impact problem (the angle equals to 90 degrees)

High velocity impact problem (the angle equals to 45 degrees)

Poisson problems f(r)= 10x8(x-2)8[36(x-1)2+2x(x-2)]y10(2-y)10+

10y8(y-2)8[36(y-1)2+2y(y-2)]x10(2-x)10

(x,y)=x10(2-x)10y10(2-y)10

Discharger(distribution of potential, 105 V)

Discharger (distribution of electric field strength, 105 V/cm)

Specifications

The source code, written in C++, has approximately 100000 lines.

Typical calculation takes from 10 minutes up to 1 hour on the SGI R10000.

RAM needed: 800 bytes per cell.

Nonstationary Preconditioned Iterative Methods for Large Linear Systems

Solving

Oleg V. Diyankov

Vladislav Y. Pravilnikov

Natalia N. Kuznetsova

Sergei V. Koshelev

Igor V. Krasnogorov

Dmitriy V. Gorshkov

Aleksey A. Nadolsky

Introduction Iterative Methods Preconditioners The PLS Code

Introduction

The work has been performed according to the contract with ExxonMobil Upstream Technology Corporation.

The main goal was to select methods which are the most appropriate ones to the ExxonMobil specific problems.

The results of many scientists and especially prof. Yousef Saad, Dr. Edmond Chow, prof. Kolotilina, Dr. Eremin have been used.

The goal of the work is to solve the linear system of equations

Ax=b, (1)

where A is a large sparse matrix, b is a right hand side vector, x is a vector of unknowns. Preconditioned Iterative Methods could be used for solving iteratively such systems.

Iterative MethodsIterative methods can be written in the general form as follows:

bxAxx

F j

j

jj

j

1

jF

j

(2)

where - the sequence of nonsingular matrices, - the sequence of real parameters.

jjj FH

bAxHxx jj

jj 1

Or, if we denote , formula (2) canbe expresses in the following form:

(3)

jHIterative methods are called stationary, if don’t depend upon the iteration count j. Otherwise,iterative methods are called nonstationary.

The most known stationary iterative methods are the Jacobi method, the Gauss-Seidel method and Successive Overrelaxation (SOR) method.The most known nonstationary iterative methods are the Conjugate Gradient (CG) method and Minimal Residual (MinRes) method and their modifications.As a rule, for nonstationary iterative methods the solution xj on J-th iteration is searched through minimization of a quadraticfunctional from xj.

bAxxx jj

jj 1

),(

),(jj

jj

jArAr

rAr

The Minimal Residual (MinRes) methodThe MinRes iterations can be written in the following form:

where

Modifications: The General Minimal Residual (GMRES) method and Quasi-Minimal Residual (QMR) method.In general the minimal residual method can be expressed in the following form:

bAxHxx jj

jj 1

where Hj is a matrix, which depends from some parameters and which is chosen to minimize |b-Axj|. This method can be applied to systems with nonsymmetrical matrix A.

The Conjugate Gradient (CG) method

jj

jj

jj

jj

j

jj

jj

jj

jj

j

jj

jj

Aprr

rr

rr

pxx

App

rr

prp

1

11

1

11

),(

),(

),(

),(

The convergence is guaranteed for systems with symmetric positive definite matrix A.Modifications: The BiConjugate Gradient (BiCG)method, The Conjugate Gradient Squared (CGS) method, The BiConjugate Gradient Stabilized (BiCGStab) method.

The solution xj on J-th iteration is constructed as an element of },...,,{ 01100 rAArrGxx i

si

so that )()( ** xxAxx ii is minimized.

jiдляrr

rr

rr

pp

rr

paxx

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pArrAprr

ji

ii

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iii

ii

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ii

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iiii

ii

0),(

),(

),(,

),(

),(

,

,

11

11

1

1111

11

The BiConjugate Gradient (BiCG) methodThe BiCG method generate two CG-type bi-orthogonal sequences of vectors, one based on A, and one on AT.

This method can be used for systems with nonsymmetrical and nonsingular matrix A.

PreconditionersbMAxM 11 (4)

The preconditioner is called incomplete, if it is constructed on a subset of indices:

0|, ijM mjiP

There is three basic type of preconditioners:• The Incomplete Cholesky (IC) factorization;• The Incomplete LU (ILU) factorization;• The Approximate Inverse (AI) factorization.

The Incomplete Cholesky (IC) factorization

ijlllal

Lal

ALLM

ii

i

kikjkjiji

i

kikiiii

T

,

,

1

1

211

1

2

The Incomplete LU (ILU) factorization

iiiiii

ii

i

kikjkjiji

mul

ijmmmam

ALUM

1

,1

1

The algorithm of the ILU(0) (ILU without extension of stencil) construction can be described as follows.

For each row of A we should fulfill the followingsteps:1. For the I-th row we select all nonzeros

having indices less than I;2. For each J we calculate aij = aij / ajj;3. For only nonzeros in I-th row with indices K

greater than I we fulfill the elimination aik = aik – aij * ajk.

There is two main strategies for construction of a preconditioners with extended stencil:• structural;• fill-in with drop tolerance.The algorithm of the FILU (fill-in ILU with drop tolerance eps ) can be described as follows.For each row of A we should fulfill the followingsteps:1. For the I-th row we select all nonzeros

having indices less than I;2. For each such J we calculate aij = aij / ajj;3. For all nonzeros in I-th and J-th rows with

indices K greater than I we fulfill the elimination aik = aik – aij * ajk if aik exists and for all other nonzero ajk we check the drop condition | aij * ajk| > eps * |Ai|.

The Approximate Inverse (AI) factorization1AM

The algorithm of the AI construction can be described as follows.

ii

ii

N

eAM

eAM

M

M

M

M

min

...2

1

For each row of matrix A we should fulfill following steps: • The determination of the fill stencil;• The calculation of the Mi coefficients from auxiliary

linear system P * mi = ai, where mi and ai consists from nonzeros of rows Mi and Ai respectively.

The PLS code

The PLS (Parallel Linear Solver) is the C++ object-oriented code for solving of a large sparse linear systems. Linear systems are considered as multi-block systems. For solving of these systems we use preconditioned iterative methods.General steps of solution process are:• Transformation of a multi-block system to a single block system;• Generation of a Preconditioner;• Execution of an Iterative Method;• Inverse transformation of obtained solution to a multi-block structure.The general object-oriented program languages concepts such as the polymorphism and the derivation has been used in the code.

Brief description of the PLS code

We’ve realized four version of the code:• Serial Unix version;• Parallel Unix version based on MPI;• Threads Unix version;• Serial Windows-NT version.The PLS code can be used both as command-line utility and as library with program interface. The user can use these interfaces in his application to choose the iterative method, the preconditioner and call solver.We’ve implemented the following iterative methods:CG, BiCG, CGS, BiCGStab, MMR(l) (the Modified Minimal Residual method),BiCGStab(L).We’ve implemented the following preconditioners: ILU, MILU, FILU, MFILU, AIP, AIP(L).Moreover, the special solver of the block Gauss-Siedel type has been realized for systems with a 3x3 block matrix.

The results, presented here, have been obtained on Pentium PC cluster system (PII/400).

Equations marked as ExxonMobil ones are samples of real equations, appearing in ExxonMobil applications. Their structure and specific characteristics is ExxonMobil proprietary information.

Numerical results

Task 1.

The solution of Poisson equation with discretization on irregular grid constructed on Voronoi cells.

Matrix size: N=50249, NNZ=349945.

Nproc Preconditioner Tprec Iterative Method Tim Ttotal

1 FILU(0.02, 0.2) 0.65 BiCGStab 16.81 17.80

1 AIP(1) 7.74 BiCGStab 39.87 47.953 ParAIP(1) 3.14 ParBiCGStab 21.35 24.497 ParAIP(1) 1.35 ParBiCGStab 10.18 11.53

Task 2ExxonMobil’s Mixed Cube problem (mc30).Matrix size: N=110700, NNZ=593100


1 FILU(0.02, 0.2) 9.79 BiCGStab 26.59 37.19

1 AIP(1) 17.81 BiCGStab 69.85 88.48

3 ParAIP(1) 7.57 ParBiCGStab 43.89 51.46

7 ParAIP(1) 3.13 ParBiCGStab 21.65 24.78

Task 3ExxonMobil’s Mixed Cube problem. (mc40)Matrix size: N=260800, NNZ=1406400


1 FILU(0.02, 0.2) 26.56 BiCGStab 102.95 131.41

1 AIP(1) 42.50 BiCGStab 237.43 281.88

3 ParAIP(1) 18.27 parBiCGStab 152.93 171.20



Task 4

ExxonMobil’s Mixed Cube problem. (mc50)Matrix size: N=507500 , NNZ=2747500 Nproc Preconditioner Tprec Iterative Method Tim Ttotal

1 FILU(0.02, 0.2) 58.86 BiCGStab 242.32 304.97

1 AIP(1) 83.78 BiCGStab 722.97 810.52




Task 5

ExxonMobil’s Mfem problem. (mfem_32_64_16)Matrix size: N=134656, NNZ=704000


1 FILU(0.02, 0.2) 2.47 BiCGStab 31.99 35.61

1 AIP(0) 2.8 BiCGStab 271.87 275.82

1 AIP(1) 20.80 BiCGStab 114.61 136.53

1 AIP(2) 261.08 BiCGStab 144.24 406.48






Task 6

ExxonMobil’s Mfem problem. (mfem_48_96_24)Matrix size: N=450432, NNZ=2395008


1 FILU(0.02, 0.2) 6.41 BiCGStab 139.16 149.33

1 AIP(1) 72.21 BiCGStab 810.89 887.03




Task 7

ExxonMobil’s Geo problem. (geo20)Matrix size: N=60983, NNZ=1430843


1 FILU(0.02, 0.2) 18.19 BiCGStab 105.48 124.89

1 FILU(0.02, 0.2) 6.21 Geo(BiCGStab) 87.71 94.35

3 ParAIP(1) 2.84 parGeo parBiCGStab

36.97 39.81


20.23 21.63


11.87 12.61

Task 8

ExxonMobil’s Geo problem. (geo30)Matrix size: N=200073, NNZ=5203713


1 FILU(0.01, 0.1) 100.23 BiCGStab 130.88 234.18

1 FILU(0.01, 0.1) 31.76 Geo(BiCGStab) 341.21 374.44

3 ParAIP(1) 10.47 parGeo(parBiCGStab) 224.51 234.98



Task 9

ExxonMobil’s Geo problem. (geo_32_64_16)Matrix size: N=244081 , NNZ=6218735


1 FILU(0.02, 0.2) 208.22 BiCGStab 125.75 337.89

1 FILU(0.02,0.2

13.74 Geo(BiCGStab) 338.18 353.52


249.75 262.40


120.34 126.25


65.41 68.81

Difference Schemes for Hyperbolic Systems Treatment

Central difference schemes for Euler gas dynamics (Kurganov-Tadmor difference schemes)

Application of KT difference scheme to gas pipe simulation problem

KT scheme for 1D Lagrangian gas dynamics KT scheme for multi-dimensional gas

dynamics irregular grid simulations

Euler equationsThe non-steady-state flow of heat-conductive viscous gas in a constant section pipe is described by system of differential equations in partial derivatives [1]:

0

z

u

t

uuDdz

dhg

z

p

z

u

t

u 2

2

eTTDdz

dhug

puu

z

u

t

4

22

22

Equation of state: ,TR

p g

TRg

1

1

8314gR (J/kg К) – universal gas constant,

- molecular weight, - Poisson index.

(1)

zt, - time and space coordinates along pipeline axis.

Tpu ,,,, - density, mass velocity, pressure, specific internal energy and gas temperature

tzTT ee , - temperature of environment (ground, water, air)

RD 2 - pipe radius

- given function, describing relief along a pipeline.

- pipeline length (simulation region).

- resistance coefficient.

- pipe diameter, R

L

zhh

)(Re, 0d

uD

Re - Reynold's number.

)(T - dynamic gas viscousity

)(00 zdd - effective rough inner pipe surface

),( tz - heat transfer coefficient

Difference schemeThe uniform grid is made along a pipeline section. Density, speed, specific internal energy and temperature of gas are defined in middle of intervals. In addition one accounting interval is attached to external borders of the pipeline, so-called fictitious cells. The known spatial distribution of appropriate values is assign in middle of intervals in the initial time moment.

Second order semi-discrete central difference scheme [2] is used for solution of equations (1):

z

tHtHtu

dt

d jjj

)()(

)( 2121 , where

22u

uu j

Difference boundary conditionsIt was supposed, that the condition of inflow gas is put on a left border, and the pressure is set on a right border. Linear dependencies of basic values chosen as initial approximation.

1. Inflow condition.),( 00 p ),( 00 TpCompositions or are set on inflow. Let's set for

definiteness, then temperature we shall express from the equation of state:),( 00 p

0

00

p

RT

g

Values with index "0" refer to border (node of accounting cell), "R" (right) - to center of the first accounting cell, "L" (left) - to center of the fictitious cell. Let's find values at the center of the fictitious cell. Linear extrapolation of density, impulse and pressure is used for this purpose.

RL 02

1)()(2)( RRL uuu

0

20

2

)(

1

2

LR

L

uppE

Accordingly, flows on boundary edge will be written down as:

001 uH

2000

2 upH

21

200

003 u

puH

The value of mass velocity is necessary for finding flows.

0 1 2 3 4 5 6 7 8 9 10

x 104

487.1506

487.1507

487.1508

487.1509

487.151

487.1511

487.1512

487.1513Ro * U

1 variant: linear extrapolation of mass velocity

1

10

)(5.0

)(5.1

R

R

R

R uuu

Small oscillation arise at such assignment order %, as it is visible from the dependence of stationary flow via coordinate, given in figure.

0u410

)( u

2 variant: linear extrapolation of impulse

0

10 2

)()(3

RR uuu

In this case flow is monotone at the left border.

0 1 2 3 4 5 6 7 8 9 10

x 104

480

482

484

486

488

490

492Ro * U, Time: 12979.166667

2. Outflow condition.Only pressure is known on outflow.0p

Values with index "0" refer to border (node of accounting cell), "L" (left) - to center of the last accounting cell, "R" (right) - to center of the fictitious cell. Let's carry out linear extrapolation of density and impulse. If we set boundary conditions and flows to

10 5.05.1 LL

10 )(5.0)(5.1)( LL uuu

01 )( uH

0

20

02 )(

u

pH

0

20

00

03

2

)(

1

)(

u

pu

H

then the large entropy trace arises on the right border (up to 1-2%), which scale does not depend on number of grid intervals.

1. Therefore we have tried to find additional difference boundary conditions being a consequence of the basic boundary condition and equations (1). For this purpose the first equation of system (1) multiply on “u” and subtract from second equation.

uuDdz

dhg

z

p

z

uu

t

u 2

Received equation multiply on , combine with the first equation (1), multiplied on and subtract the third equation. In result we have,

u

22u

2

2uu

Ddz

dhgu

z

up

z

u

t

Subtract last equation from the first equation (1) multiplied on :

2

2uu

Ddz

dhgu

z

up

zu

t

Multiply this equation by and combine with the first equation (1) multiplied by :

p

p

22

2uu

Ddz

dhgu

p

z

uc

z

pu

t

p

, where c is a sound speed

The equation below is received using the equation of state of ideal gas

2

2)1())(1( uu

Ddz

dhgu

z

up

z

pu

t

p

Last equation write down in difference form and take into account, that at ,

follow , then

Lz 0

t

p

2

2)1())(1( uu

Ddz

dhgu

z

up

z

pu

(2)

The law of entropy conservation is used as third difference boundary condition:

02

t

p

tt

sT

The law of entropy conservation in finite differences has the following form:

uuDdz

dhg

z

p

z 22

3. Entropy equation

(3)

Using (2) and (3), one can obtain a system of equations with respect to and 0 0u

08

1

022

1

1

1

822

200

200000

000

2000

302

uuuuD

uuppuupp

ppuuuu

D

pp

LLLLLLL

LLLL

L

L

L

L

Using the found values and , it is possible to find the other unknowns0 0u

21

2000

0

upE

LR uuu 02

LR 02

RRR uu )(

LR EEE 02

001 uH

2000

2 upH

21

200

003 u

puH

Error in temperature at the right boundary decreases with the increasing of grid points number.

Numerical resultsThe test task given in the report [1], we solve for evaluation of the numerical technique. Gas compressibility, heat exchange with walls, friction on internal surface of a pipe, viscosity essentially influence gas dynamics in these tasks.

Stationary gas flow in heat-insulated pipeline100N - number of difference intervals. Calculations were carried out in interval Lz ,0

F ix e d v a lu e s : 1D , 510L , 273eT , 210 .

B o u n d a r y c o n d i t io n s : 6106 Leftp , 273LeftT , 6105.1 Rightp .

In i t ia l c o n d i t io n s : l in e a r in te rp o la t io n b e tw e e n th e f o l lo w in g v a lu e s o n b o u n d a r y

6106 Leftp a n d 6105.1 Rightp ; 8.42Left a n d 4/LeftRight ;

5.11Leftu a n d LeftRight uu 4 . T e m p e ra tu r e i s c o n s ta n t : 273oT .

The stationary task 1 was solved by a method of establishment flow under given boundary conditions.

Heat exchange with an environment is completely absent in this task: . Calculation results in comparison by exact solutions are given in Fig. 1. Good consent is observed with exact solutions.

0

0 1 2 3 4 5 6 7 8 9 10

x 104

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6x 10

6

p

z0 1 2 3 4 5 6 7 8 9 10

x 104

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

T-2

73

z

0 1 2 3 4 5 6 7 8 9 10

x 104

10

15

20

25

30

35

40

45

50

u

z0 1 2 3 4 5 6 7 8 9 10

x 104

487.12

487.122

487.124

487.126

487.128

487.13

487.132

487.134

487.136

487.138

487.14

Ro

* u

z

Pressure Temperature

Velocity Mass flow

Fig. 1. Stationary gas flow distribution of pressure, temperature, velocity andmass flow in in heat-insulated pipeline. Solid line – numerical results,stars – exact solutions.

In this task the convergence of numerical solution to analytical was analyzed.

101

102

103

10-3

10-2

10-1

100

Con

verg

ence

Amount of spatial cells is plotted on the blue diagram on X-axis in logarithmic scale, logarithm of norm of a difference numerical solutions and analytical – on Y-axis.

The green colour plots the cubic law: .3

30

0 x

xyy

It is possible to make a conclusion that the solution converges to analytical with the third order depending on amount of spatial cells.

Literature

1. Anuchin M.G., Dremov V.V. Model for calculate flow of natural gas on gas pipeline. RFNC-VNIITF report, № ПС.99.7432, Snezhinsk, 1999.2. Kurganov A., Tadmor E. High-resolution central schemes. UCLA Report №16, 1999.

3D Elastic-Plastic Modeling of Processes of Ceramics

Formation Problem Statement Material Model (by Bychenkov) Lagrange Irregular Voronoi Grids Central Difference Scheme for Irregular

Grids in 2D and 3D Cases

Problem Statement

Model of Elastic-Plastic Deformations I

SSlS

x

gSugpt

gE

gSgpt

gu

t

g

s

iTi

ii

ii

ii

2

u

)(u

)()(

0

Model of Elastic-Plastic Deformations II

x

u

y

u

SSS

SS

SS

yx

xxyyxy

xyyy

xyxx

)(2

2

2

Model of Elastic-Plastic Deformations III

PK

J

c

P

PlsE

P

tcP

psp

p

pijij

2

0

1

2

2

The additional modes of the model

Porosity accounting Material crash accounting Welding of the material accounting

The KT Difference Scheme for Lagrangian Gas Dynamics

ux

u

t

g

upug

p

t

ugt

g

0

0))2

((

0

0

2

Difference scheme for 1D Lagrangian Gas Dynamics I

0)(1

0)(1

0)(1

0

12/1

2/12/1

2/12/1

gk

gk

k

Ek

Ek

k

Ik

Ik

k

k

FFht

g

FFht

E

FFht

It

m

Difference scheme for 1D Lagrangian Gas Dynamics II

)(5.0))()((5.0

)(5.0))()((5.0

)(5.0)(5.0

2/12/1111

2/12/12/12/12/1

2/12/12/12/12/1

kkkkg

k

kkkkI

k

kkkkI

k

ggcuuF

EEcupupF

IIcppF

Difference scheme for 1D Lagrangian Gas Dynamics III

2/12/1

2/12/1

2/1

112/1

2/1

112/1

2

2

2

2

2

2

kkk

kkk

kkk

kkk

kkk

kkk

Dgh

gg

Dgh

gg

DEh

EE

DEh

EE

DIh

II

DIh

II

Difference scheme for 1D Lagrangian Gas Dynamics IV

),mod(min

),mod(min

),mod(min

2/12/12/12/32/1

11

11

h

gg

h

ggDg

h

EE

h

EEDE

h

II

h

IIDI

kkkkk

kkkkk

kkkkk

Difference scheme for multidimensional irregular grid

The multidimensional Voronoi Grid is used To obtain scheme for multi-dimensional

case one should reformulate original 1D KT scheme to make generalization more obvious (see the next 2 slides)

Difference scheme for 1D Lagrangian Gas Dynamics (Variation for generalization for

multidimensional irregular grid)

))(2

1(

2

)(2

))(2

1(

2

)(2

))()((5.0

))(2

1(

2

)(2

)(5.0

2/12/32/12/3

2/12/311

11

112/1

11

112/1

kkkk

kkkg

k

kkkk

kkkkE

k

kkkk

kkkkI

k

DgDgggc

DuDuh

uF

DEDEEEc

DpuDpuh

upupF

DIDIIIc

DpDph

ppF

Difference scheme for 1D Lagrangian Gas Dynamics (Variation for generalization for

multidimensional irregular grid)

)))(,)mod((min))(,)mod(((min2

1)(

)2()(

))()((2

)(2

112

11

22/12/1

kkkkk

kkkk

kkk

kkkk

xxxxxD

xxxx

xDxg

cDuDu

hu

t

x

A test problem for the lagrangian difference scheme

In the next five slides the results of a test problem numerical modeling are presented

The initial conditions are: the first region – density 4, specific internal energy – 0.5, velocity – 1; the second region – density – 1, specific internal energy – 1e-6, velocity – 0.

Boundary conditions: the left boundary – piston with velocity 1, the right one – rigid wall.

The plots present density and velocity for 10 successive times.

density

velocity

density

velocity

density

velocity

density

velocity

density

velocity

Numerical results discussion

The numerical results show, that the scheme has inherited its good quality from the Euler parent KT scheme

The smoothness is minimal (even after more, than 10 shock reflections the number of point at the shock is not more then five)

Some entropy traces are presented (at the boundaries and at the place, where initially the shock has been located)

Explanation of some details of the multidimensional scheme, presented on the

next slide. is a set of numbers of cells, which are

neighbor cells to the k-th one DI, DE are the results of minmod procedure

applied to the gradients (in common case six ones), calculated in the Delauneux triangles, including k-th point

Dr is a result of minmod operation applied to the corresponding laplacians, calculated in the neighbor points

k

Lagrangian KT Scheme for Irregular Grid

k k

k k k j jkj

j k j k jkj

k k k j jkj

j k j k jkj

k k

j k

g

t

tI g p p n

cI I DI DI n

tE g pu pu n

cE E DE DE n

tr u

gr Dr

rc

r r

k k

k k

0

1

2 2

1

2

1

2 2

1

2

1

2

( ) ( ) ( ( ))

( ) (( ) ( ) ) ( ( ))

( )

( )

n jkj k

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Nonclassified activity of CML RFNC-VNIITF Oleg V. Diyankov Head of Computational MHD Laboratory.

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